Spreading of Perturbations in Long-Range Interacting Classical Lattice Models

Lieb-Robinson-type bounds are reported for a large class of classical Hamiltonian lattice models. By a suitable rescaling of energy or time, such bounds can be constructed for interactions of arbitrarily long range. The bound quantifies the dependence of the system’s dynamics on a perturbation of the initial state. The effect of the perturbation is found to be effectively restricted to the interior of a causal region of logarithmic shape, with only small, algebraically decaying effects in the exterior. A refined bound, sharper than conventional Lieb-Robinson bounds, is required to correctly capture the shape of the causal region, as confirmed by numerical results for classical long-range $XY$ chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.

Figure 1

Illustration of the spatiotemporal behavior of perturbations. The contour plots show $\ln {\mathcal{Q}}_{ij}^{pq}$ as a function of the distance $|i-j|$ and time $t$, for chains of length $N=256$. Left: for the $XY$ chain with nearest-neighbor interactions, the effect of a perturbation is restricted to the interior of a cone-shaped region. Right: for the $\alpha XY$ chain with $\alpha =1/2$, the contours spread faster than linearly in space, as expected from (9), illustrating supersonic propagation.

Figure 2

Numerical data and fits of the spreading of perturbations in the $\alpha XY$ chain with $\alpha =1/2$. The contour plots show $\ln {\mathcal{Q}}_{ij}^{pq}$ as a function of the distance $|i-j|$ on a logarithmic scale and time $t$ on a linear scale. Left: numerical data for a chain of length $N=4096$. Center: fit of the function $c{B}_{ij}^{pq}(t/z)$ [based on the weaker bound in (5c)] to the numerical data of ${\mathcal{Q}}_{ij}^{pq}(t)$, with fit parameters $c=0.0064$ and $z=1.47$, yielding a residual sum of squares of 0.157. The contours of the bound are approximately linear for large $|i-j|$ and $t$, but this does not correctly capture the actual behavior of the data. Right: As in the center plot, but fitting the parameters $\tilde{c}=21.5$ and $\tilde{z}=11.2$ in the stronger bound (11) and yielding a residual sum of squares as small as 0.0065.

Figure 3

Left and center: system-size dependence of the parameters $\tilde{c}$ and $\tilde{z}$ when fitting (11) to the numerical data of $\alpha XY$ chains, using initial conditions with zero initial momenta. The data for $\alpha =1/2$ in the left plot are well described by the power laws $\tilde{c}\approx 2.3N^{b_{\tilde{c}}}$ and $\tilde{z}\approx 1.2N^{b_{\tilde{z}}}$ with $b_{\tilde{c}}\approx b_{\tilde{z}}\approx 0.27$. Right: $b_{\tilde{c}}$ and $b_{\tilde{z}}$ as functions of the exponent $\alpha$. For $\alpha <1$ both are well fitted by the linear function $(1-\alpha )/2$. Right: short-time behavior of the difference quotients ${\mathcal{Q}}_{ij}^{qq}$, ${\mathcal{Q}}_{ij}^{qp}$, ${\mathcal{Q}}_{ij}^{pq}$, and ${\mathcal{Q}}_{ij}^{qq}$, plotted on a log-log scale. Data are for $\alpha =1/2$ and chain length $N=4096$. The solid data curves display a linear, quadratic, or cubic initial growth, in agreement with the corresponding bounds. Dotted lines are fits of $c{B}_{ij}(t/z)$, with $c$ and $z$ as fitting parameters.